Optimal. Leaf size=110 \[ -\frac {2 \csc (a+b x)}{5 b d (d \tan (a+b x))^{3/2}}+\frac {6 \cos (a+b x)}{5 b d^2 \sqrt {d \tan (a+b x)}}+\frac {6 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{5 b d^2 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}} \]
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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 = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2677, 2681,
2650, 2652, 2719} \begin {gather*} \frac {6 \cos (a+b x)}{5 b d^2 \sqrt {d \tan (a+b x)}}+\frac {6 \sin (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{5 b d^2 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}-\frac {2 \csc (a+b x)}{5 b d (d \tan (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2650
Rule 2652
Rule 2677
Rule 2681
Rule 2719
Rubi steps
\begin {align*} \int \frac {\csc (a+b x)}{(d \tan (a+b x))^{5/2}} \, dx &=-\frac {2 \csc (a+b x)}{5 b d (d \tan (a+b x))^{3/2}}-\frac {3 \int \frac {\csc (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx}{5 d^2}\\ &=-\frac {2 \csc (a+b x)}{5 b d (d \tan (a+b x))^{3/2}}-\frac {\left (3 \sqrt {\sin (a+b x)}\right ) \int \frac {\sqrt {\cos (a+b x)}}{\sin ^{\frac {3}{2}}(a+b x)} \, dx}{5 d^2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\\ &=-\frac {2 \csc (a+b x)}{5 b d (d \tan (a+b x))^{3/2}}+\frac {6 \cos (a+b x)}{5 b d^2 \sqrt {d \tan (a+b x)}}+\frac {\left (6 \sqrt {\sin (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)} \, dx}{5 d^2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\\ &=-\frac {2 \csc (a+b x)}{5 b d (d \tan (a+b x))^{3/2}}+\frac {6 \cos (a+b x)}{5 b d^2 \sqrt {d \tan (a+b x)}}+\frac {(6 \sin (a+b x)) \int \sqrt {\sin (2 a+2 b x)} \, dx}{5 d^2 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}\\ &=-\frac {2 \csc (a+b x)}{5 b d (d \tan (a+b x))^{3/2}}+\frac {6 \cos (a+b x)}{5 b d^2 \sqrt {d \tan (a+b x)}}+\frac {6 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{5 b d^2 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 1.85, size = 105, normalized size = 0.95 \begin {gather*} \frac {2 \left (2 \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\tan ^2(a+b x)\right ) \sec ^2(a+b x)-\left (3-4 \csc ^2(a+b x)+\csc ^4(a+b x)\right ) \sqrt {\sec ^2(a+b x)}\right ) \sin (a+b x) \sqrt {d \tan (a+b x)}}{5 b d^3 \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. \(963\) vs.
\(2(121)=242\).
time = 0.38, size = 964, normalized size = 8.76
method | result | size |
default | \(\text {Expression too large to display}\) | \(964\) |
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(-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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc {\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {5}{2}}}\, dx \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 = -0.01 \begin {gather*} \int \frac {1}{\sin \left (a+b\,x\right )\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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