Optimal. Leaf size=223 \[ -\frac {3 \left (a^2+6 a b+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{8 \sqrt {a} f}+\frac {3 \sqrt {b} (a+b) \tanh ^{-1}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{2 f}+\frac {3 (a+3 b) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {3 (a+b) \csc ^2(e+f x) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {\cot (e+f x) \csc ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{4 f} \]
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Rubi [A]
time = 0.23, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3745, 478, 591,
596, 537, 223, 212, 385, 213} \begin {gather*} -\frac {3 \left (a^2+6 a b+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )}{8 \sqrt {a} f}+\frac {3 (a+3 b) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{8 f}-\frac {3 (a+b) \csc ^2(e+f x) \sec (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{8 f}+\frac {3 \sqrt {b} (a+b) \tanh ^{-1}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )}{2 f}-\frac {\cot (e+f x) \csc ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{4 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 213
Rule 223
Rule 385
Rule 478
Rule 537
Rule 591
Rule 596
Rule 3745
Rubi steps
\begin {align*} \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx &=\frac {\text {Subst}\left (\int \frac {x^4 \left (a-b+b x^2\right )^{3/2}}{\left (-1+x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {\cot (e+f x) \csc ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{4 f}+\frac {\text {Subst}\left (\int \frac {x^2 \sqrt {a-b+b x^2} \left (3 (a-b)+6 b x^2\right )}{\left (-1+x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{4 f}\\ &=-\frac {3 (a+b) \csc ^2(e+f x) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {\cot (e+f x) \csc ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{4 f}+\frac {\text {Subst}\left (\int \frac {x^2 \left (3 (a-b) (a+5 b)+6 b (a+3 b) x^2\right )}{\left (-1+x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{8 f}\\ &=\frac {3 (a+3 b) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {3 (a+b) \csc ^2(e+f x) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {\cot (e+f x) \csc ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{4 f}-\frac {\text {Subst}\left (\int \frac {-6 (a-b) b (a+3 b)-24 b^2 (a+b) x^2}{\left (-1+x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{16 b f}\\ &=\frac {3 (a+3 b) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {3 (a+b) \csc ^2(e+f x) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {\cot (e+f x) \csc ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{4 f}+\frac {(3 b (a+b)) \text {Subst}\left (\int \frac {1}{\sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{2 f}+\frac {\left (3 \left (a^2+6 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{8 f}\\ &=\frac {3 (a+3 b) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {3 (a+b) \csc ^2(e+f x) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {\cot (e+f x) \csc ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{4 f}+\frac {(3 b (a+b)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{2 f}+\frac {\left (3 \left (a^2+6 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1+a x^2} \, dx,x,\frac {\sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{8 f}\\ &=-\frac {3 \left (a^2+6 a b+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{8 \sqrt {a} f}+\frac {3 \sqrt {b} (a+b) \tanh ^{-1}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{2 f}+\frac {3 (a+3 b) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {3 (a+b) \csc ^2(e+f x) \sec (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 f}-\frac {\cot (e+f x) \csc ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{4 f}\\ \end {align*}
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Mathematica [A]
time = 4.74, size = 415, normalized size = 1.86 \begin {gather*} \frac {\cos (e+f x) \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)} \left (-2 \csc ^2(e+f x) \left (3 a+5 b+2 a \csc ^2(e+f x)\right )+8 b \sec ^2(e+f x)+\frac {3 \left (16 \sqrt {a} \sqrt {b} (a+b) \tanh ^{-1}\left (\frac {-\sqrt {a} \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}{2 \sqrt {b}}\right )+\left (a^2+6 a b+b^2\right ) \left (2 \tanh ^{-1}\left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-\frac {\sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}{\sqrt {a}}\right )+\log \left (a-2 b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+\sqrt {a} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}\right )\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\cos ^2(e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right )}}{\sqrt {a} \sqrt {\left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}\right )}{16 \sqrt {2} f} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(6193\) vs.
\(2(195)=390\).
time = 0.33, size = 6194, normalized size = 27.78
method | result | size |
default | \(\text {Expression too large to display}\) | \(6194\) |
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 [A]
time = 4.12, size = 1449, normalized size = 6.50 \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(-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 = -0.00 \begin {gather*} \int \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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