Optimal. Leaf size=312 \[ -\frac {a^{5/2} (c+d)^2 \left (3 c^2-26 c d+163 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{64 d^{5/2} f}-\frac {a^3 (c+d) \left (3 c^2-26 c d+163 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{64 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^3 \left (3 c^2-26 c d+163 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {a^3 (3 c-17 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f} \]
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Rubi [A]
time = 0.47, antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2842, 3060,
2849, 2854, 211} \begin {gather*} -\frac {a^{5/2} (c+d)^2 \left (3 c^2-26 c d+163 d^2\right ) \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^{5/2} f}-\frac {a^3 \left (3 c^2-26 c d+163 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {a^3 (c+d) \left (3 c^2-26 c d+163 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{64 d^2 f \sqrt {a \sin (e+f x)+a}}+\frac {a^3 (3 c-17 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}{4 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 2842
Rule 2849
Rule 2854
Rule 3060
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2} \, dx &=-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac {\int \sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a^2 (c+13 d)-\frac {1}{2} a^2 (3 c-17 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2} \, dx}{4 d}\\ &=\frac {a^3 (3 c-17 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac {\left (a^2 \left (3 c^2-26 c d+163 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx}{48 d^2}\\ &=-\frac {a^3 \left (3 c^2-26 c d+163 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {a^3 (3 c-17 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac {\left (a^2 (c+d) \left (3 c^2-26 c d+163 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx}{64 d^2}\\ &=-\frac {a^3 (c+d) \left (3 c^2-26 c d+163 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{64 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^3 \left (3 c^2-26 c d+163 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {a^3 (3 c-17 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac {\left (a^2 (c+d)^2 \left (3 c^2-26 c d+163 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{128 d^2}\\ &=-\frac {a^3 (c+d) \left (3 c^2-26 c d+163 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{64 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^3 \left (3 c^2-26 c d+163 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {a^3 (3 c-17 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}-\frac {\left (a^3 (c+d)^2 \left (3 c^2-26 c d+163 d^2\right )\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^2 f}\\ &=-\frac {a^{5/2} (c+d)^2 \left (3 c^2-26 c d+163 d^2\right ) \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^{5/2} f}-\frac {a^3 (c+d) \left (3 c^2-26 c d+163 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{64 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^3 \left (3 c^2-26 c d+163 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {a^3 (3 c-17 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\\ \end {align*}
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Mathematica [A]
time = 1.17, size = 327, normalized size = 1.05 \begin {gather*} \frac {(a (1+\sin (e+f x)))^{5/2} \left (\frac {(c+d)^2 \left (3 c^2-26 c d+163 d^2\right ) \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^{5/2}}+\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 (9 c^3-63 c^2 d-773 c d^2-581 d^3+4 d^2 (9 c+23 d) \cos (2 (e+f x))-2 d \left (3 c^2+158 c d+181 d^2\right ) \sin (e+f x)+12 d^3 \sin (3 (e+f x))\right )}{3 d^2}\right )}{128 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 668 vs.
\(2 (289) = 578\).
time = 1.12, size = 1809, normalized size = 5.80 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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