Optimal. Leaf size=112 \[ -\frac {7 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d^3}-\frac {7 \cos (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{2 b d^2 \sqrt {\sin (2 a+2 b x)}}-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}} \]
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Rubi [A] time = 0.15, antiderivative size = 112, 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 = {2609, 2612, 2615, 2572, 2639} \[ -\frac {7 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d^3}-\frac {7 \cos (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{2 b d^2 \sqrt {\sin (2 a+2 b x)}}-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2572
Rule 2609
Rule 2612
Rule 2615
Rule 2639
Rubi steps
\begin {align*} \int \frac {\cos ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {7 \int \cos ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx}{d^2}\\ &=-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {7 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d^3}-\frac {7 \int \cos (a+b x) \sqrt {d \tan (a+b x)} \, dx}{2 d^2}\\ &=-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {7 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d^3}-\frac {\left (7 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)} \, dx}{2 d^2 \sqrt {\sin (a+b x)}}\\ &=-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {7 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d^3}-\frac {\left (7 \cos (a+b x) \sqrt {d \tan (a+b x)}\right ) \int \sqrt {\sin (2 a+2 b x)} \, dx}{2 d^2 \sqrt {\sin (2 a+2 b x)}}\\ &=-\frac {2 \cos ^3(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {7 \cos (a+b x) E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {d \tan (a+b x)}}{2 b d^2 \sqrt {\sin (2 a+2 b x)}}-\frac {7 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b d^3}\\ \end {align*}
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Mathematica [C] time = 0.57, size = 77, normalized size = 0.69 \[ \frac {\sin (a+b x) \left (-14 \tan ^2(a+b x) \sqrt {\sec ^2(a+b x)} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\tan ^2(a+b x)\right )+\cos (2 (a+b x))-13\right )}{6 b (d \tan (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \tan \left (b x + a\right )} \cos \left (b x + a\right )^{3}}{d^{2} \tan \left (b x + a\right )^{2}}, x\right ) \]
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 \[ \int \frac {\cos \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.52, size = 523, normalized size = 4.67 \[ \frac {\left (2 \left (\cos ^{4}\left (b x +a \right )\right ) \sqrt {2}+42 \EllipticE \left (\sqrt {-\frac {-\sin \left (b x +a \right )-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \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 {-\sin \left (b x +a \right )-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}-21 \EllipticF \left (\sqrt {-\frac {-\sin \left (b x +a \right )-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \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 {-\sin \left (b x +a \right )-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}+42 \EllipticE \left (\sqrt {-\frac {-\sin \left (b x +a \right )-1+\cos \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 {-\sin \left (b x +a \right )-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}-21 \EllipticF \left (\sqrt {-\frac {-\sin \left (b x +a \right )-1+\cos \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 {-\sin \left (b x +a \right )-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}+7 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}-21 \cos \left (b x +a \right ) \sqrt {2}\right ) \sin \left (b x +a \right ) \sqrt {2}}{12 b \cos \left (b x +a \right )^{2} \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {\cos \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (a+b\,x\right )}^3}{{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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