Optimal. Leaf size=242 \[ \frac {10 a e^4 \left (11 a^2+6 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{231 d \sqrt {e \sin (c+d x)}}-\frac {10 a e^3 \left (11 a^2+6 b^2\right ) \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}-\frac {2 a e \left (11 a^2+6 b^2\right ) \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{143 d e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.32, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2692, 2862, 2669, 2635, 2642, 2641} \[ -\frac {10 a e^3 \left (11 a^2+6 b^2\right ) \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}+\frac {10 a e^4 \left (11 a^2+6 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{231 d \sqrt {e \sin (c+d x)}}+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}-\frac {2 a e \left (11 a^2+6 b^2\right ) \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{143 d e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2635
Rule 2641
Rule 2642
Rule 2669
Rule 2692
Rule 2862
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2} \, dx &=\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac {2}{13} \int (a+b \cos (c+d x)) \left (\frac {13 a^2}{2}+2 b^2+\frac {17}{2} a b \cos (c+d x)\right ) (e \sin (c+d x))^{7/2} \, dx\\ &=\frac {34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac {4}{143} \int \left (\frac {13}{4} a \left (11 a^2+6 b^2\right )+\frac {1}{4} b \left (177 a^2+44 b^2\right ) \cos (c+d x)\right ) (e \sin (c+d x))^{7/2} \, dx\\ &=\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac {34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac {1}{11} \left (a \left (11 a^2+6 b^2\right )\right ) \int (e \sin (c+d x))^{7/2} \, dx\\ &=-\frac {2 a \left (11 a^2+6 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac {34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac {1}{77} \left (5 a \left (11 a^2+6 b^2\right ) e^2\right ) \int (e \sin (c+d x))^{3/2} \, dx\\ &=-\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 a \left (11 a^2+6 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac {34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac {1}{231} \left (5 a \left (11 a^2+6 b^2\right ) e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx\\ &=-\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 a \left (11 a^2+6 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac {34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac {\left (5 a \left (11 a^2+6 b^2\right ) e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{231 \sqrt {e \sin (c+d x)}}\\ &=\frac {10 a \left (11 a^2+6 b^2\right ) e^4 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{231 d \sqrt {e \sin (c+d x)}}-\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 a \left (11 a^2+6 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac {34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.49, size = 205, normalized size = 0.85 \[ \frac {(e \sin (c+d x))^{7/2} \left (154 b \left (78 a^2+11 b^2\right ) \csc ^3(c+d x)-\frac {2080 a \left (11 a^2+6 b^2\right ) F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )}{\sin ^{\frac {7}{2}}(c+d x)}+\frac {1}{3} \csc ^3(c+d x) \left (-77 b \left (624 a^2+73 b^2\right ) \cos (2 (c+d x))-154 b \left (b^2-78 a^2\right ) \cos (4 (c+d x))-156 a \left (506 a^2+213 b^2\right ) \cos (c+d x)+234 a \left (44 a^2-39 b^2\right ) \cos (3 (c+d x))+4914 a b^2 \cos (5 (c+d x))+693 b^3 \cos (6 (c+d x))\right )\right )}{48048 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (b^{3} e^{3} \cos \left (d x + c\right )^{5} + 3 \, a b^{2} e^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{2} b e^{3} \cos \left (d x + c\right ) + {\left (3 \, a^{2} b - b^{3}\right )} e^{3} \cos \left (d x + c\right )^{3} - a^{3} e^{3} + {\left (a^{3} - 3 \, a b^{2}\right )} e^{3} \cos \left (d x + c\right )^{2}\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.32, size = 252, normalized size = 1.04 \[ \frac {\frac {2 b \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}} \left (9 \left (\cos ^{2}\left (d x +c \right )\right ) b^{2}+39 a^{2}+4 b^{2}\right )}{117 e}-\frac {e^{4} a \left (-126 b^{2} \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )+\left (-66 a^{2}+216 b^{2}\right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (176 a^{2}-30 b^{2}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+55 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}+30 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}\right )}{231 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________