Integrand size = 25, antiderivative size = 596 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3} \, dx=\frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{13/4} d e^{3/2}}-\frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{13/4} d e^{3/2}}-\frac {a \left (8 a^2+37 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 \left (a^2-b^2\right )^3 d e^2 \sqrt {\cos (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{8 \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{8 \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}+\frac {b}{2 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)}} \] Output:
5/8*b^(3/2)*(7*a^2+2*b^2)*arctan(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^( 1/4)/e^(1/2))/(-a^2+b^2)^(13/4)/d/e^(3/2)-5/8*b^(3/2)*(7*a^2+2*b^2)*arctan h(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/(-a^2+b^2)^(13/4) /d/e^(3/2)-1/4*a*(8*a^2+37*b^2)*(e*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d*x +1/2*c),2^(1/2))/(a^2-b^2)^3/d/e^2/cos(d*x+c)^(1/2)-5/8*a*b*(7*a^2+2*b^2)* cos(d*x+c)^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(b-(-a^2+b^2)^(1/2)),2^ (1/2))/(a^2-b^2)^3/(b-(-a^2+b^2)^(1/2))/d/e/(e*cos(d*x+c))^(1/2)-5/8*a*b*( 7*a^2+2*b^2)*cos(d*x+c)^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(b+(-a^2+b ^2)^(1/2)),2^(1/2))/(a^2-b^2)^3/(b+(-a^2+b^2)^(1/2))/d/e/(e*cos(d*x+c))^(1 /2)+1/2*b/(a^2-b^2)/d/e/(e*cos(d*x+c))^(1/2)/(a+b*sin(d*x+c))^2+9/4*a*b/(a ^2-b^2)^2/d/e/(e*cos(d*x+c))^(1/2)/(a+b*sin(d*x+c))-1/4*(5*b*(7*a^2+2*b^2) -a*(8*a^2+37*b^2)*sin(d*x+c))/(a^2-b^2)^3/d/e/(e*cos(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.68 (sec) , antiderivative size = 922, normalized size of antiderivative = 1.55 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3),x]
Output:
-1/8*(Cos[c + d*x]^(3/2)*((-2*(8*a^4 + 72*a^2*b^2 + 10*b^4)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos [c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I /8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d* x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b ^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^( 1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((8 *a^3*b + 37*a*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/ 4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b ]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*S qrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt [b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/((a - b)^3*(a + b)^3*d*(e*Cos[c + d*x])^(3/2)) + (Cos[ c + d*x]^2*(-1/2*(b^3*Cos[c + d*x])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^2) - (13*a*b^3*Cos[c + d*x])/(4*(a^2 - b^2)^3*(a + b*Sin[c + d*x])) + (2*...
Time = 2.78 (sec) , antiderivative size = 560, normalized size of antiderivative = 0.94, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.960, Rules used = {3042, 3173, 27, 3042, 3343, 27, 3042, 3345, 27, 3042, 3346, 3042, 3121, 3042, 3119, 3180, 266, 827, 218, 221, 3042, 3286, 3042, 3284}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}dx\) |
\(\Big \downarrow \) 3173 |
\(\displaystyle \frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}-\frac {\int -\frac {4 a-5 b \sin (c+d x)}{2 (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {4 a-5 b \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}dx}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {4 a-5 b \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}dx}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3343 |
\(\displaystyle \frac {\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {\int -\frac {8 a^2-27 b \sin (c+d x) a+10 b^2}{2 (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}dx}{a^2-b^2}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \frac {2 \left (4 a^2+5 b^2\right )-27 a b \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}dx}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {2 \left (4 a^2+5 b^2\right )-27 a b \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}dx}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3345 |
\(\displaystyle \frac {\frac {-\frac {2 \int \frac {\sqrt {e \cos (c+d x)} \left (8 a^4+72 b^2 a^2+b \left (8 a^2+37 b^2\right ) \sin (c+d x) a+10 b^4\right )}{2 (a+b \sin (c+d x))}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {-\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (2 \left (4 a^4+36 b^2 a^2+5 b^4\right )+a b \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {-\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (2 \left (4 a^4+36 b^2 a^2+5 b^4\right )+a b \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3346 |
\(\displaystyle \frac {\frac {-\frac {a \left (8 a^2+37 b^2\right ) \int \sqrt {e \cos (c+d x)}dx+5 b^2 \left (7 a^2+2 b^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {-\frac {a \left (8 a^2+37 b^2\right ) \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 b^2 \left (7 a^2+2 b^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {\frac {-\frac {\frac {a \left (8 a^2+37 b^2\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{\sqrt {\cos (c+d x)}}+5 b^2 \left (7 a^2+2 b^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx+\frac {a \left (8 a^2+37 b^2\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\frac {-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3180 |
\(\displaystyle \frac {\frac {-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (\frac {b e \int \frac {\sqrt {e \cos (c+d x)}}{b^2 \cos ^2(c+d x) e^2+\left (a^2-b^2\right ) e^2}d(e \cos (c+d x))}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\frac {-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (\frac {2 b e \int \frac {e^2 \cos ^2(c+d x)}{b^2 e^4 \cos ^4(c+d x)+\left (a^2-b^2\right ) e^2}d\sqrt {e \cos (c+d x)}}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {\frac {-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (\frac {2 b e \left (\frac {\int \frac {1}{b e^2 \cos ^2(c+d x)+\sqrt {b^2-a^2} e}d\sqrt {e \cos (c+d x)}}{2 b}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \cos ^2(c+d x)}d\sqrt {e \cos (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\frac {-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \cos ^2(c+d x)}d\sqrt {e \cos (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (-\frac {a e \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \sin \left (c+d x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 b}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3286 |
\(\displaystyle \frac {\frac {-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (-\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (-\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \sin \left (c+d x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {\frac {-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}+\frac {a e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{b d \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{b d \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}}{2 \left (a^2-b^2\right )}+\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}}{4 \left (a^2-b^2\right )}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}\) |
Input:
Int[1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3),x]
Output:
b/(2*(a^2 - b^2)*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2) + ((9*a* b)/((a^2 - b^2)*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])) + (-(((2*a* (8*a^2 + 37*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[C os[c + d*x]]) + 5*b^2*(7*a^2 + 2*b^2)*((2*b*e*(ArcTan[(Sqrt[b]*Sqrt[e]*Cos [c + d*x])/(-a^2 + b^2)^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4)*Sqrt[e]) - Ar cTanh[(Sqrt[b]*Sqrt[e]*Cos[c + d*x])/(-a^2 + b^2)^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4)*Sqrt[e])))/d + (a*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos [c + d*x]]) + (a*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^ 2]), (c + d*x)/2, 2])/(b*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]])))/ ((a^2 - b^2)*e^2)) - (2*(5*b*(7*a^2 + 2*b^2) - a*(8*a^2 + 37*b^2)*Sin[c + d*x]))/((a^2 - b^2)*d*e*Sqrt[e*Cos[c + d*x]]))/(2*(a^2 - b^2)))/(4*(a^2 - b^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b ^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_ )]), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[a*(g/(2*b)) Int[1/(S qrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (-Simp[a*(g/(2*b)) In t[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x], x] + Simp[b*(g/f) Su bst[Int[Sqrt[x]/(g^2*(a^2 - b^2) + b^2*x^2), x], x, g*Cos[e + f*x]], x])] / ; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) Int[(g*Cos[e + f*x])^p *(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ [a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt Q[p, -1] && IntegerQ[2*m]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((c_.) + (d_.)*sin[(e_.) + (f_.)* (x_)]))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d/b Int [(g*Cos[e + f*x])^p, x], x] + Simp[(b*c - a*d)/b Int[(g*Cos[e + f*x])^p/( a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 11.71 (sec) , antiderivative size = 3539, normalized size of antiderivative = 5.94
Input:
int(1/(e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
(-4/e*b*(3/64/(a+b)^3/(a-b)^3/(e^2*(a^2-b^2)/b^2)^(1/4)*(16*(2*e*cos(1/2*d *x+1/2*c)^2-e)^(1/2)*(e^2*(a^2-b^2)/b^2)^(1/4)*(cos(1/2*d*x+1/2*c)^2-1/2)* b^2+e*(ln((2*e*cos(1/2*d*x+1/2*c)^2-e-(e^2*(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1 /2*d*x+1/2*c)^2-e)^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2))/(2*e*cos(1/2*d *x+1/2*c)^2-e+(e^2*(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2) *2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2)))+2*arctan((2^(1/2)*(2*e*cos(1/2*d*x+1/ 2*c)^2-e)^(1/2)-(e^2*(a^2-b^2)/b^2)^(1/4))/(e^2*(a^2-b^2)/b^2)^(1/4))+2*ar ctan((2^(1/2)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/4) )/(e^2*(a^2-b^2)/b^2)^(1/4)))*(4*b^2*cos(1/2*d*x+1/2*c)^4-4*b^2*cos(1/2*d* x+1/2*c)^2+a^2)*2^(1/2))*(a^2+1/3*b^2)/e/(4*b^2*cos(1/2*d*x+1/2*c)^4-4*b^2 *cos(1/2*d*x+1/2*c)^2+a^2)+1/16/(a-b)^3/(a+b)^3*(3*a^2+b^2)/(e^2*(a^2-b^2) /b^2)^(1/4)*2^(1/2)*(ln((2*e*cos(1/2*d*x+1/2*c)^2-e-(e^2*(a^2-b^2)/b^2)^(1 /4)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2))/ (2*e*cos(1/2*d*x+1/2*c)^2-e+(e^2*(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2 *c)^2-e)^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2)))+2*arctan(2^(1/2)/(e^2*( a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)+1)-2*arctan(-2^(1/2 )/(e^2*(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)+1))+1/8*(3* a^2+b^2)/(a^2-b^2)^3/e*2^(1/2)/(cos(1/2*d*x+1/2*c)-1/2*2^(1/2))*(-2*sin(1/ 2*d*x+1/2*c)^2*e+e)^(1/2)-1/8*(3*a^2+b^2)/(a^2-b^2)^3/e*2^(1/2)/(cos(1/2*d *x+1/2*c)+1/2*2^(1/2))*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)+5/128*a^2/(a...
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(1/(e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^3,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(1/(e*cos(d*x+c))**(3/2)/(a+b*sin(d*x+c))**3,x)
Output:
Timed out
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(1/(e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^3,x, algorithm="maxima")
Output:
Timed out
\[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:
integrate(1/(e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
integrate(1/((e*cos(d*x + c))^(3/2)*(b*sin(d*x + c) + a)^3), x)
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3} \,d x \] Input:
int(1/((e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^3),x)
Output:
int(1/((e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^3), x)
\[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right ) a^{2} b +\cos \left (d x +c \right )^{2} a^{3}}d x \right )}{e^{2}} \] Input:
int(1/(e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^3,x)
Output:
(sqrt(e)*int(sqrt(cos(c + d*x))/(cos(c + d*x)**2*sin(c + d*x)**3*b**3 + 3* cos(c + d*x)**2*sin(c + d*x)**2*a*b**2 + 3*cos(c + d*x)**2*sin(c + d*x)*a* *2*b + cos(c + d*x)**2*a**3),x))/e**2