Optimal. Leaf size=110 \[ \frac {7 d^3 \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}-\frac {7 d^2 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{2 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b} \]
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Rubi [A]
time = 0.10, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2674, 2678,
2681, 2652, 2719} \begin {gather*} \frac {7 d^3 \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}-\frac {7 d^2 \sin (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2652
Rule 2674
Rule 2678
Rule 2681
Rule 2719
Rubi steps
\begin {align*} \int \sin ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx &=\frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-\left (7 d^2\right ) \int \frac {\sin ^3(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\\ &=\frac {7 d^3 \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}+\frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-\frac {1}{2} \left (7 d^2\right ) \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\\ &=\frac {7 d^3 \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}+\frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-\frac {\left (7 d^2 \sqrt {\sin (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)} \, dx}{2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\\ &=\frac {7 d^3 \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}+\frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}-\frac {\left (7 d^2 \sin (a+b x)\right ) \int \sqrt {\sin (2 a+2 b x)} \, dx}{2 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}\\ &=\frac {7 d^3 \sin ^3(a+b x)}{3 b (d \tan (a+b x))^{3/2}}-\frac {7 d^2 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{2 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \sin ^3(a+b x) \sqrt {d \tan (a+b x)}}{b}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 5.44, size = 90, normalized size = 0.82 \begin {gather*} \frac {\left (-28 \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\tan ^2(a+b x)\right ) \sec (a+b x)+2 \cos (a+b x) (13+\cos (2 (a+b x))) \sqrt {\sec ^2(a+b x)}\right ) (d \tan (a+b x))^{3/2}}{12 b \sqrt {\sec ^2(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(539\) vs.
\(2(123)=246\).
time = 0.30, size = 540, normalized size = 4.91
method | result | size |
default | \(-\frac {\left (-1+\cos \left (b x +a \right )\right )^{2} \left (2 \left (\cos ^{4}\left (b x +a \right )\right ) \sqrt {2}+21 \cos \left (b x +a \right ) \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}-42 \cos \left (b x +a \right ) \EllipticE \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}+21 \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}-42 \EllipticE \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}-11 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}+21 \cos \left (b x +a \right ) \sqrt {2}-12 \sqrt {2}\right ) \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \cos \left (b x +a \right ) \left (\cos \left (b x +a \right )+1\right )^{2} \sqrt {2}}{12 b \sin \left (b x +a \right )^{6}}\) | \(540\) |
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 [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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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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.01 \begin {gather*} \int {\sin \left (a+b\,x\right )}^3\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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