Optimal. Leaf size=285 \[ \frac {5 a^{3/2} (c-15 d) (c+d)^3 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{64 d^{3/2} f}+\frac {5 a^2 (c-15 d) (c+d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{64 d f \sqrt {a+a \sin (e+f x)}}+\frac {5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a+a \sin (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.38, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2842, 21, 2849,
2854, 211} \begin {gather*} \frac {5 a^{3/2} (c-15 d) (c+d)^3 \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{64 d^{3/2} f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a \sin (e+f x)+a}}+\frac {a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt {a \sin (e+f x)+a}}+\frac {5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt {a \sin (e+f x)+a}}+\frac {5 a^2 (c-15 d) (c+d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{64 d f \sqrt {a \sin (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 211
Rule 2842
Rule 2849
Rule 2854
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx &=-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a+a \sin (e+f x)}}+\frac {\int \frac {\left (-\frac {1}{2} a^2 (c-15 d)-\frac {1}{2} a^2 (c-15 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{4 d}\\ &=-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a+a \sin (e+f x)}}-\frac {(a (c-15 d)) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx}{8 d}\\ &=\frac {a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a+a \sin (e+f x)}}-\frac {(5 a (c-15 d) (c+d)) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx}{48 d}\\ &=\frac {5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a+a \sin (e+f x)}}-\frac {\left (5 a (c-15 d) (c+d)^2\right ) \int \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx}{64 d}\\ &=\frac {5 a^2 (c-15 d) (c+d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{64 d f \sqrt {a+a \sin (e+f x)}}+\frac {5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a+a \sin (e+f x)}}-\frac {\left (5 a (c-15 d) (c+d)^3\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{128 d}\\ &=\frac {5 a^2 (c-15 d) (c+d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{64 d f \sqrt {a+a \sin (e+f x)}}+\frac {5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a+a \sin (e+f x)}}+\frac {\left (5 a^2 (c-15 d) (c+d)^3\right ) \text {Subst}\left (\int \frac {1}{a+d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{64 d f}\\ &=\frac {5 a^{3/2} (c-15 d) (c+d)^3 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{64 d^{3/2} f}+\frac {5 a^2 (c-15 d) (c+d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{64 d f \sqrt {a+a \sin (e+f x)}}+\frac {5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.95, size = 318, normalized size = 1.12 \begin {gather*} \frac {(a (1+\sin (e+f x)))^{3/2} \left (-\frac {5 (c-15 d) (c+d)^3 \left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right )-\log \left (\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\sqrt {c+d \sin (e+f x)}\right )\right )}{d^{3/2}}-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)} \left (15 c^3+455 c^2 d+653 c d^2+285 d^3-4 d^2 (17 c+15 d) \cos (2 (e+f x))+2 d \left (59 c^2+190 c d+93 d^2\right ) \sin (e+f x)-12 d^3 \sin (3 (e+f x))\right )}{3 d}\right )}{128 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 582 vs.
\(2 (262) = 524\).
time = 1.11, size = 1637, normalized size = 5.74 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________