Integrand size = 25, antiderivative size = 444 \[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \sec (c+d x)} \, dx=-\frac {b \sqrt [4]{a^2-b^2} e^{3/2} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{5/2} d}-\frac {b \sqrt [4]{a^2-b^2} e^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{5/2} d}+\frac {2 \left (a^2-3 b^2\right ) e^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a^3 d \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (a^2-b^2\right ) e^2 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^3 \left (a^2-b^2-a \sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (a^2-b^2\right ) e^2 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^3 \left (a^2-b^2+a \sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 e (3 b-a \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 a^2 d} \]
-b*(a^2-b^2)^(1/4)*e^(3/2)*arctan(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^( 1/4)/e^(1/2))/a^(5/2)/d-b*(a^2-b^2)^(1/4)*e^(3/2)*arctanh(a^(1/2)*(e*sin(d *x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))/a^(5/2)/d-2/3*(a^2-3*b^2)*e^2*(sin(1 /2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticF(cos(1/2* c+1/4*Pi+1/2*d*x),2^(1/2))*sin(d*x+c)^(1/2)/a^3/d/(e*sin(d*x+c))^(1/2)-b^2 *(a^2-b^2)*e^2*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d* x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*a/(a-(a^2-b^2)^(1/2)),2^(1/2))*s in(d*x+c)^(1/2)/a^3/d/(a^2-b^2-a*(a^2-b^2)^(1/2))/(e*sin(d*x+c))^(1/2)-b^2 *(a^2-b^2)*e^2*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d* x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*a/(a+(a^2-b^2)^(1/2)),2^(1/2))*s in(d*x+c)^(1/2)/a^3/d/(a^2-b^2+a*(a^2-b^2)^(1/2))/(e*sin(d*x+c))^(1/2)+2/3 *e*(3*b-a*cos(d*x+c))*(e*sin(d*x+c))^(1/2)/a^2/d
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 24.66 (sec) , antiderivative size = 1959, normalized size of antiderivative = 4.41 \[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \sec (c+d x)} \, dx =\text {Too large to display} \]
(-2*(b + a*Cos[c + d*x])*Csc[c + d*x]*(e*Sin[c + d*x])^(3/2))/(3*a*d*(a + b*Sec[c + d*x])) + ((b + a*Cos[c + d*x])*Sec[c + d*x]*(e*Sin[c + d*x])^(3/ 2)*((4*a*Cos[c + d*x]^2*(b + a*Sqrt[1 - Sin[c + d*x]^2])*((b*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin [c + d*x]] + a*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(3/4)) - (5 *a*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d* x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^ 2 - b^2)] + 2*(2*a^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[ c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) - (2*b* Cos[c + d*x]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[a]*(2*Ar cTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTa n[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + Log[Sqrt[a ^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin [c + d*x]] - Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sq...
Time = 2.03 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.98, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.920, Rules used = {3042, 4360, 25, 25, 3042, 3344, 27, 3042, 3346, 3042, 3121, 3042, 3120, 3181, 25, 266, 756, 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.
| \(\displaystyle \int \frac {(e \sin (c+d x))^{3/2}}{a+b \sec (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2}}{a-b \csc \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\cos (c+d x) (e \sin (c+d x))^{3/2}}{-a \cos (c+d x)-b}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cos (c+d x) (e \sin (c+d x))^{3/2}}{b+a \cos (c+d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\cos (c+d x) (e \sin (c+d x))^{3/2}}{a \cos (c+d x)+b}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (-e \cos \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}{a \sin \left (c+d x+\frac {\pi }{2}\right )+b}dx\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {2 e^2 \int -\frac {2 a b-\left (a^2-3 b^2\right ) \cos (c+d x)}{2 (b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}}dx}{3 a^2}+\frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \int \frac {2 a b-\left (a^2-3 b^2\right ) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}}dx}{3 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \int \frac {2 a b-\left (3 b^2-a^2\right ) \sin \left (c+d x-\frac {\pi }{2}\right )}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{3 a^2}\) |
| \(\Big \downarrow \) 3346 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \int \frac {1}{(b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}}dx}{a}-\frac {\left (a^2-3 b^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}}dx}{a}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{a}-\frac {\left (a^2-3 b^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}}dx}{a}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{a}-\frac {\left (a^2-3 b^2\right ) \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{a \sqrt {e \sin (c+d x)}}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{a}-\frac {\left (a^2-3 b^2\right ) \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{a \sqrt {e \sin (c+d x)}}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{a}-\frac {2 \left (a^2-3 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \sqrt {e \sin (c+d x)}}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 3181 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \left (-\frac {a e \int -\frac {1}{\sqrt {e \sin (c+d x)} \left (\left (a^2-b^2\right ) e^2-a^2 e^2 \sin ^2(c+d x)\right )}d(e \sin (c+d x))}{d}-\frac {b \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 \sqrt {a^2-b^2}}-\frac {b \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 \sqrt {a^2-b^2}}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \sqrt {e \sin (c+d x)}}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \left (\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\left (a^2-b^2\right ) e^2-a^2 e^2 \sin ^2(c+d x)\right )}d(e \sin (c+d x))}{d}-\frac {b \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 \sqrt {a^2-b^2}}-\frac {b \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 \sqrt {a^2-b^2}}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \sqrt {e \sin (c+d x)}}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \left (\frac {2 a e \int \frac {1}{\left (a^2-b^2\right ) e^2-a^2 e^4 \sin ^4(c+d x)}d\sqrt {e \sin (c+d x)}}{d}-\frac {b \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 \sqrt {a^2-b^2}}-\frac {b \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 \sqrt {a^2-b^2}}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \sqrt {e \sin (c+d x)}}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \left (\frac {2 a e \left (\frac {\int \frac {1}{\sqrt {a^2-b^2} e-a e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 e \sqrt {a^2-b^2}}+\frac {\int \frac {1}{a e^2 \sin ^2(c+d x)+\sqrt {a^2-b^2} e}d\sqrt {e \sin (c+d x)}}{2 e \sqrt {a^2-b^2}}\right )}{d}-\frac {b \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 \sqrt {a^2-b^2}}-\frac {b \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 \sqrt {a^2-b^2}}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \sqrt {e \sin (c+d x)}}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \left (\frac {2 a e \left (\frac {\int \frac {1}{\sqrt {a^2-b^2} e-a e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 e \sqrt {a^2-b^2}}+\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 \sqrt {a} e^{3/2} \left (a^2-b^2\right )^{3/4}}\right )}{d}-\frac {b \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 \sqrt {a^2-b^2}}-\frac {b \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 \sqrt {a^2-b^2}}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \sqrt {e \sin (c+d x)}}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \left (-\frac {b \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 \sqrt {a^2-b^2}}-\frac {b \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 \sqrt {a^2-b^2}}+\frac {2 a e \left (\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 \sqrt {a} e^{3/2} \left (a^2-b^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 \sqrt {a} e^{3/2} \left (a^2-b^2\right )^{3/4}}\right )}{d}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \sqrt {e \sin (c+d x)}}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \left (-\frac {b \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 \sqrt {a^2-b^2}}-\frac {b \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 \sqrt {a^2-b^2}}+\frac {2 a e \left (\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 \sqrt {a} e^{3/2} \left (a^2-b^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 \sqrt {a} e^{3/2} \left (a^2-b^2\right )^{3/4}}\right )}{d}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \sqrt {e \sin (c+d x)}}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \left (-\frac {b \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 \sqrt {a^2-b^2} \sqrt {e \sin (c+d x)}}-\frac {b \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 \sqrt {a^2-b^2} \sqrt {e \sin (c+d x)}}+\frac {2 a e \left (\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 \sqrt {a} e^{3/2} \left (a^2-b^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 \sqrt {a} e^{3/2} \left (a^2-b^2\right )^{3/4}}\right )}{d}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \sqrt {e \sin (c+d x)}}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \left (-\frac {b \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 \sqrt {a^2-b^2} \sqrt {e \sin (c+d x)}}-\frac {b \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 \sqrt {a^2-b^2} \sqrt {e \sin (c+d x)}}+\frac {2 a e \left (\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 \sqrt {a} e^{3/2} \left (a^2-b^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 \sqrt {a} e^{3/2} \left (a^2-b^2\right )^{3/4}}\right )}{d}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \sqrt {e \sin (c+d x)}}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {2 e \sqrt {e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac {e^2 \left (\frac {3 b \left (a^2-b^2\right ) \left (\frac {2 a e \left (\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 \sqrt {a} e^{3/2} \left (a^2-b^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 \sqrt {a} e^{3/2} \left (a^2-b^2\right )^{3/4}}\right )}{d}+\frac {b \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {a^2-b^2} \left (a-\sqrt {a^2-b^2}\right ) \sqrt {e \sin (c+d x)}}-\frac {b \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {a^2-b^2} \left (\sqrt {a^2-b^2}+a\right ) \sqrt {e \sin (c+d x)}}\right )}{a}-\frac {2 \left (a^2-3 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \sqrt {e \sin (c+d x)}}\right )}{3 a^2}\) |
(2*e*(3*b - a*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*a^2*d) - (e^2*((-2*(a ^2 - 3*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a*d*Sqrt [e*Sin[c + d*x]]) + (3*b*(a^2 - b^2)*((2*a*e*(ArcTan[(Sqrt[a]*Sqrt[e]*Sin[ c + d*x])/(a^2 - b^2)^(1/4)]/(2*Sqrt[a]*(a^2 - b^2)^(3/4)*e^(3/2)) + ArcTa nh[(Sqrt[a]*Sqrt[e]*Sin[c + d*x])/(a^2 - b^2)^(1/4)]/(2*Sqrt[a]*(a^2 - b^2 )^(3/4)*e^(3/2))))/d + (b*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/ 2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(Sqrt[a^2 - b^2]*(a - Sqrt[a^2 - b^2])* d*Sqrt[e*Sin[c + d*x]]) - (b*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(Sqrt[a^2 - b^2]*(a + Sqrt[a^2 - b^2 ])*d*Sqrt[e*Sin[c + d*x]])))/a))/(3*a^2)
3.3.35.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]*((a_) + (b_.)*sin[(e_.) + (f_.)* (x_)])), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[-a/(2*q) Int[1/( Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (Simp[b*(g/f) Subst[ Int[1/(Sqrt[x]*(g^2*(a^2 - b^2) + b^2*x^2)), x], x, g*Cos[e + f*x]], x] - S imp[a/(2*q) Int[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), 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[g*(g* Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d* p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Simp[g^2*( (p - 1)/(b^2*(m + p)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Si n[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^ 2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1 , 0] && 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]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 10.10 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.43
| method | result | size |
| default | \(\frac {\frac {e b \left (-\left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{a^{2}}\right )^{\frac {1}{4}} \ln \left (\frac {\sqrt {e \sin \left (d x +c \right )}+\left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{a^{2}}\right )^{\frac {1}{4}}}{\sqrt {e \sin \left (d x +c \right )}-\left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{a^{2}}\right )^{\frac {1}{4}}}\right )-2 \left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{a^{2}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\left (\frac {a^{2} e^{2}-b^{2} e^{2}}{a^{2}}\right )^{\frac {1}{4}}}\right )+4 \sqrt {e \sin \left (d x +c \right )}\right )}{2 a^{2}}+\frac {\sqrt {\cos \left (d x +c \right )^{2} e \sin \left (d x +c \right )}\, a \,e^{2} \left (-\frac {\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+2 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )}{3 a^{2} \sqrt {\cos \left (d x +c \right )^{2} e \sin \left (d x +c \right )}}+\frac {b^{2} \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )}{a^{4} \sqrt {\cos \left (d x +c \right )^{2} e \sin \left (d x +c \right )}}-\frac {b^{2} \left (a^{2}-b^{2}\right ) \left (-\frac {\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {1}{1-\frac {\sqrt {a^{2}-b^{2}}}{a}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a^{2}-b^{2}}\, a \sqrt {\cos \left (d x +c \right )^{2} e \sin \left (d x +c \right )}\, \left (1-\frac {\sqrt {a^{2}-b^{2}}}{a}\right )}+\frac {\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {1}{1+\frac {\sqrt {a^{2}-b^{2}}}{a}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a^{2}-b^{2}}\, a \sqrt {\cos \left (d x +c \right )^{2} e \sin \left (d x +c \right )}\, \left (1+\frac {\sqrt {a^{2}-b^{2}}}{a}\right )}\right )}{a^{4}}\right )}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(636\) |
(1/2*e*b*(-(e^2*(a^2-b^2)/a^2)^(1/4)*ln(((e*sin(d*x+c))^(1/2)+(e^2*(a^2-b^ 2)/a^2)^(1/4))/((e*sin(d*x+c))^(1/2)-(e^2*(a^2-b^2)/a^2)^(1/4)))-2*(e^2*(a ^2-b^2)/a^2)^(1/4)*arctan((e*sin(d*x+c))^(1/2)/((a^2*e^2-b^2*e^2)/a^2)^(1/ 4))+4*(e*sin(d*x+c))^(1/2))/a^2+(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*a*e^2*(- 1/3/a^2/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*((-sin(d*x+c)+1)^(1/2)*(2*sin(d* x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2) )+2*cos(d*x+c)^2*sin(d*x+c))+b^2/a^4*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2 )^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*EllipticF((-sin (d*x+c)+1)^(1/2),1/2*2^(1/2))-b^2*(a^2-b^2)/a^4*(-1/2/(a^2-b^2)^(1/2)/a*(- sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2* e*sin(d*x+c))^(1/2)/(1-(a^2-b^2)^(1/2)/a)*EllipticPi((-sin(d*x+c)+1)^(1/2) ,1/(1-(a^2-b^2)^(1/2)/a),1/2*2^(1/2))+1/2/(a^2-b^2)^(1/2)/a*(-sin(d*x+c)+1 )^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c) )^(1/2)/(1+(a^2-b^2)^(1/2)/a)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1+(a^2-b ^2)^(1/2)/a),1/2*2^(1/2))))/cos(d*x+c)/(e*sin(d*x+c))^(1/2))/d
Timed out. \[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \sec (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \sec (c+d x)} \, dx=\int \frac {\left (e \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}{a + b \sec {\left (c + d x \right )}}\, dx \]
\[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \sec (c+d x)} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}}{b \sec \left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \sec (c+d x)} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}}{b \sec \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}}{b+a\,\cos \left (c+d\,x\right )} \,d x \]