Optimal. Leaf size=150 \[ -\frac {(i a+b) (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(i a-b) (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 (b c+a d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b (c+d \tan (e+f x))^{3/2}}{3 f} \]
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Rubi [A]
time = 0.22, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3609, 3620,
3618, 65, 214} \begin {gather*} \frac {2 (a d+b c) \sqrt {c+d \tan (e+f x)}}{f}-\frac {(b+i a) (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(-b+i a) (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 b (c+d \tan (e+f x))^{3/2}}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 3609
Rule 3618
Rule 3620
Rubi steps
\begin {align*} \int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx &=\frac {2 b (c+d \tan (e+f x))^{3/2}}{3 f}+\int \sqrt {c+d \tan (e+f x)} (a c-b d+(b c+a d) \tan (e+f x)) \, dx\\ &=\frac {2 (b c+a d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b (c+d \tan (e+f x))^{3/2}}{3 f}+\int \frac {-2 b c d+a \left (c^2-d^2\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {2 (b c+a d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {1}{2} \left ((a-i b) (c-i d)^2\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} \left ((a+i b) (c+i d)^2\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {2 (b c+a d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {\left ((i a+b) (c-i d)^2\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 f}-\frac {\left ((i a-b) (c+i d)^2\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f}\\ &=\frac {2 (b c+a d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {\left ((a-i b) (c-i d)^2\right ) \text {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}-\frac {\left ((a+i b) (c+i d)^2\right ) \text {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}\\ &=-\frac {(i a+b) (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(i a-b) (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 (b c+a d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b (c+d \tan (e+f x))^{3/2}}{3 f}\\ \end {align*}
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Mathematica [A]
time = 0.56, size = 140, normalized size = 0.93 \begin {gather*} \frac {-3 i (a-i b) (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+3 i (a+i b) (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )+2 \sqrt {c+d \tan (e+f x)} (4 b c+3 a d+b d \tan (e+f x))}{3 f} \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. \(917\) vs.
\(2(126)=252\).
time = 0.42, size = 918, normalized size = 6.12
method | result | size |
derivativedivides | \(\frac {\frac {2 b \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 a d \sqrt {c +d \tan \left (f x +e \right )}+2 b c \sqrt {c +d \tan \left (f x +e \right )}+\frac {\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c +\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,c^{2}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,d^{2}-2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b c d \right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (-2 \sqrt {c^{2}+d^{2}}\, a \,d^{2}-2 \sqrt {c^{2}+d^{2}}\, b c d -\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c +\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,c^{2}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,d^{2}-2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b c d \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 d}+\frac {\frac {\left (\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c -\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,c^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,d^{2}+2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b c d \right ) \ln \left (d \tan \left (f x +e \right )+c -\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (-2 \sqrt {c^{2}+d^{2}}\, a \,d^{2}-2 \sqrt {c^{2}+d^{2}}\, b c d +\frac {\left (\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c -\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,c^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,d^{2}+2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b c d \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 d}}{f}\) | \(918\) |
default | \(\frac {\frac {2 b \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 a d \sqrt {c +d \tan \left (f x +e \right )}+2 b c \sqrt {c +d \tan \left (f x +e \right )}+\frac {\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c +\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,c^{2}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,d^{2}-2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b c d \right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (-2 \sqrt {c^{2}+d^{2}}\, a \,d^{2}-2 \sqrt {c^{2}+d^{2}}\, b c d -\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c +\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,c^{2}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,d^{2}-2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b c d \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 d}+\frac {\frac {\left (\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c -\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,c^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,d^{2}+2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b c d \right ) \ln \left (d \tan \left (f x +e \right )+c -\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (-2 \sqrt {c^{2}+d^{2}}\, a \,d^{2}-2 \sqrt {c^{2}+d^{2}}\, b c d +\frac {\left (\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c -\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,c^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a \,d^{2}+2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b c d \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 d}}{f}\) | \(918\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 \left (a + b \tan {\left (e + f x \right )}\right ) \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \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 [B]
time = 17.41, size = 2823, normalized size = 18.82 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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