Integrand size = 25, antiderivative size = 430 \[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {a} b \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\left (a^2-b^2\right )^{5/4} d e^{3/2}}-\frac {\sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\left (a^2-b^2\right )^{5/4} d e^{3/2}}+\frac {2 (b-a \cos (c+d x))}{\left (a^2-b^2\right ) d e \sqrt {e \sin (c+d x)}}-\frac {b^2 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{\left (a^2-b^2\right ) \left (a-\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {b^2 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{\left (a^2-b^2\right ) \left (a+\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {2 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{\left (a^2-b^2\right ) d e^2 \sqrt {\sin (c+d x)}} \]
b*arctan(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))*a^(1/2)/(a^ 2-b^2)^(5/4)/d/e^(3/2)-b*arctanh(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1 /4)/e^(1/2))*a^(1/2)/(a^2-b^2)^(5/4)/d/e^(3/2)+2*(b-a*cos(d*x+c))/(a^2-b^2 )/d/e/(e*sin(d*x+c))^(1/2)+b^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))*sin(d*x+c)^(1/2)/(a^2-b^2)/d/e/(a-(a^2-b^2)^(1/2))/(e*sin(d *x+c))^(1/2)+b^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)) *sin(d*x+c)^(1/2)/(a^2-b^2)/d/e/(a+(a^2-b^2)^(1/2))/(e*sin(d*x+c))^(1/2)+2 *a*(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)*EllipticE (cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*sin(d*x+c))^(1/2)/(a^2-b^2)/d/e^2/s in(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 13.80 (sec) , antiderivative size = 711, normalized size of antiderivative = 1.65 \[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx=\frac {(b+a \cos (c+d x)) \left (-\frac {\sqrt {a} \left (b+a \sqrt {\cos ^2(c+d x)}\right ) \sec ^2(c+d x) \sin ^{\frac {3}{2}}(c+d x) \left (\cos (c+d x) \left (3 \sqrt {2} b \left (-a^2+b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-\sqrt {2} \sqrt {a} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+a \sin (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+\sqrt {2} \sqrt {a} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+a \sin (c+d x)\right )\right )+8 a^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right )+(2+2 i) b \sqrt {\cos ^2(c+d x)} \left (3 \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {(1+i) \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-(1+i) \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+i a \sin (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+(1+i) \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+i a \sin (c+d x)\right )\right )-(4-4 i) \sqrt {a} b \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right )\right )}{(a-b) (a+b) (b+a \cos (c+d x))}+24 (b-a \cos (c+d x)) \tan (c+d x)\right )}{12 \left (a^2-b^2\right ) d (a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}} \]
((b + a*Cos[c + d*x])*(-((Sqrt[a]*(b + a*Sqrt[Cos[c + d*x]^2])*Sec[c + d*x ]^2*Sin[c + d*x]^(3/2)*(Cos[c + d*x]*(3*Sqrt[2]*b*(-a^2 + b^2)^(3/4)*(2*Ar cTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcT an[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*S in[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*S qrt[Sin[c + d*x]] + a*Sin[c + d*x]]) + 8*a^(5/2)*AppellF1[3/4, -1/2, 1, 7/ 4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2)) + (2 + 2*I)*b*Sqrt[Cos[c + d*x]^2]*(3*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[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]] + L og[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]]) - (4 - 4*I)*Sqrt[a]*b*AppellF1[3/4, 1/2, 1, 7/4, Sin[ c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))))/((a - b)*(a + b)*(b + a*Cos[c + d*x]))) + 24*(b - a*Cos[c + d*x])*Tan[c + d*x])) /(12*(a^2 - b^2)*d*(a + b*Sec[c + d*x])*(e*Sin[c + d*x])^(3/2))
Time = 1.97 (sec) , antiderivative size = 404, 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, 4360, 25, 25, 3042, 25, 3345, 27, 3042, 3346, 3042, 3121, 3042, 3119, 3180, 25, 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.
| \(\displaystyle \int \frac {1}{(e \sin (c+d x))^{3/2} (a+b \sec (c+d x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2} \left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle -\int -\frac {\cos (c+d x)}{(b+a \cos (c+d x)) (e \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle \int -\frac {\sin \left (c+d x-\frac {\pi }{2}\right )}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2} \left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\left (e \cos \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^{3/2} \left (b-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )}dx\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {2 \int \frac {\left (\cos (c+d x) a^2+2 b a\right ) \sqrt {e \sin (c+d x)}}{2 (b+a \cos (c+d x))}dx}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {\int \frac {\left (\cos (c+d x) a^2+2 b a\right ) \sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)}dx}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {\int \frac {\sqrt {-e \cos \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a^2+2 b a\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3346 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \int \frac {\sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)}dx+a \int \sqrt {e \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{b-a \sin \left (c+d x-\frac {\pi }{2}\right )}dx+a \int \sqrt {e \sin (c+d x)}dx}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{b-a \sin \left (c+d x-\frac {\pi }{2}\right )}dx+\frac {a \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{b-a \sin \left (c+d x-\frac {\pi }{2}\right )}dx+\frac {a \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{b-a \sin \left (c+d x-\frac {\pi }{2}\right )}dx+\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3180 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \left (-\frac {a e \int -\frac {\sqrt {e \sin (c+d x)}}{\left (a^2-b^2\right ) e^2-a^2 e^2 \sin ^2(c+d x)}d(e \sin (c+d x))}{d}-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a}+\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a}\right )+\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \left (\frac {a e \int \frac {\sqrt {e \sin (c+d x)}}{\left (a^2-b^2\right ) e^2-a^2 e^2 \sin ^2(c+d x)}d(e \sin (c+d x))}{d}-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a}+\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a}\right )+\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \left (\frac {2 a e \int \frac {e^2 \sin ^2(c+d x)}{\left (a^2-b^2\right ) e^2-a^2 e^4 \sin ^4(c+d x)}d\sqrt {e \sin (c+d x)}}{d}-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a}+\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a}\right )+\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \left (\frac {2 a e \left (\frac {\int \frac {1}{\sqrt {a^2-b^2} e-a e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 a}-\frac {\int \frac {1}{a e^2 \sin ^2(c+d x)+\sqrt {a^2-b^2} e}d\sqrt {e \sin (c+d x)}}{2 a}\right )}{d}-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a}+\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a}\right )+\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \left (-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a}+\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a}+\frac {2 a e \left (\frac {\int \frac {1}{\sqrt {a^2-b^2} e-a e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 a}-\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{d}\right )+\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \left (-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a}+\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a}+\frac {2 a e \left (\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}-\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{d}\right )+\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \left (-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a}+\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a}+\frac {2 a e \left (\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}-\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{d}\right )+\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \left (-\frac {b e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a \sqrt {e \sin (c+d x)}}+\frac {b e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a \sqrt {e \sin (c+d x)}}+\frac {2 a e \left (\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}-\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{d}\right )+\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \left (-\frac {b e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a \sqrt {e \sin (c+d x)}}+\frac {b e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a \sqrt {e \sin (c+d x)}}+\frac {2 a e \left (\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}-\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{d}\right )+\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a b \left (\frac {b e \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \left (a-\sqrt {a^2-b^2}\right ) \sqrt {e \sin (c+d x)}}+\frac {b e \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \left (\sqrt {a^2-b^2}+a\right ) \sqrt {e \sin (c+d x)}}+\frac {2 a e \left (\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}-\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{d}\right )+\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}\) |
(2*(b - a*Cos[c + d*x]))/((a^2 - b^2)*d*e*Sqrt[e*Sin[c + d*x]]) - ((2*a*El lipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]] ) + a*b*((2*a*e*(-1/2*ArcTan[(Sqrt[a]*Sqrt[e]*Sin[c + d*x])/(a^2 - b^2)^(1 /4)]/(a^(3/2)*(a^2 - b^2)^(1/4)*Sqrt[e]) + ArcTanh[(Sqrt[a]*Sqrt[e]*Sin[c + d*x])/(a^2 - b^2)^(1/4)]/(2*a^(3/2)*(a^2 - b^2)^(1/4)*Sqrt[e])))/d + (b* e*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[ c + d*x]])/(a*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (b*e*Ellipti cPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]] )/(a*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]])))/((a^2 - b^2)*e^2)
3.3.38.3.1 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[(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[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[(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]
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 = 8.16 (sec) , antiderivative size = 661, normalized size of antiderivative = 1.54
| method | result | size |
| default | \(\frac {b e \left (\frac {2 \arctan \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}}}\right )-\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 e^{2} \left (a -b \right ) \left (a +b \right ) \left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{a^{2}}\right )^{\frac {1}{4}}}+\frac {2}{e^{2} \left (a^{2}-b^{2}\right ) \sqrt {e \sin \left (d x +c \right )}}\right )-\frac {b^{2} \left (\sqrt {a^{2}-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 {EllipticPi}\left (\sqrt {-\sin \left (d x +c \right )+1}, -\frac {a}{\sqrt {a^{2}-b^{2}}-a}, \frac {\sqrt {2}}{2}\right )-\sqrt {a^{2}-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 {EllipticPi}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {a}{a +\sqrt {a^{2}-b^{2}}}, \frac {\sqrt {2}}{2}\right )+4 \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 {EllipticE}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a -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 +\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, a \operatorname {EllipticPi}\left (\sqrt {-\sin \left (d x +c \right )+1}, -\frac {a}{\sqrt {a^{2}-b^{2}}-a}, \frac {\sqrt {2}}{2}\right )+\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, a \operatorname {EllipticPi}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {a}{a +\sqrt {a^{2}-b^{2}}}, \frac {\sqrt {2}}{2}\right )-4 \cos \left (d x +c \right )^{2} a \right )}{2 e \left (a +\sqrt {a^{2}-b^{2}}\right ) \left (\sqrt {a^{2}-b^{2}}-a \right ) \left (a +b \right ) \left (a -b \right ) \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(661\) |
(b*e*(1/2/e^2/(a-b)/(a+b)/(e^2*(a^2-b^2)/a^2)^(1/4)*(2*arctan((e*sin(d*x+c ))^(1/2)/(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)/(e*sin(d*x+c))^(1/2))-1/2*b^2*((a^2-b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2 )*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticPi((-sin(d*x+c)+1)^(1/2) ,-a/((a^2-b^2)^(1/2)-a),1/2*2^(1/2))-(a^2-b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2) *(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticPi((-sin(d*x+c)+1)^(1/2), a/(a+(a^2-b^2)^(1/2)),1/2*2^(1/2))+4*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2 )^(1/2)*sin(d*x+c)^(1/2)*EllipticE((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*a-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))*a+(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^ (1/2)*sin(d*x+c)^(1/2)*a*EllipticPi((-sin(d*x+c)+1)^(1/2),-a/((a^2-b^2)^(1 /2)-a),1/2*2^(1/2))+(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c )^(1/2)*a*EllipticPi((-sin(d*x+c)+1)^(1/2),a/(a+(a^2-b^2)^(1/2)),1/2*2^(1/ 2))-4*cos(d*x+c)^2*a)/e/(a+(a^2-b^2)^(1/2))/((a^2-b^2)^(1/2)-a)/(a+b)/(a-b )/cos(d*x+c)/(e*sin(d*x+c))^(1/2))/d
Timed out. \[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx=\int \frac {1}{\left (e \sin {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (a + b \sec {\left (c + d x \right )}\right )}\, dx \]
Timed out. \[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}\,\left (b+a\,\cos \left (c+d\,x\right )\right )} \,d x \]