Optimal. Leaf size=333 \[ -\frac {2 (2 a+b) \tanh (e+f x)}{3 f (a-b)^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {b (5 a+3 b) \sinh (e+f x) \cosh (e+f x)}{3 f (a-b)^3 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\tanh (e+f x) \text {sech}^2(e+f x)}{3 f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {8 \sqrt {a} \sqrt {b} (a+b) \cosh (e+f x) E\left (\tan ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 f (a-b)^4 \sqrt {a+b \sinh ^2(e+f x)} \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}+\frac {(3 a+b) (a+3 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 a f (a-b)^4 \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.40, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3196, 470, 527, 525, 418, 411} \[ -\frac {2 (2 a+b) \tanh (e+f x)}{3 f (a-b)^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {b (5 a+3 b) \sinh (e+f x) \cosh (e+f x)}{3 f (a-b)^3 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\tanh (e+f x) \text {sech}^2(e+f x)}{3 f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {8 \sqrt {a} \sqrt {b} (a+b) \cosh (e+f x) E\left (\tan ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 f (a-b)^4 \sqrt {a+b \sinh ^2(e+f x)} \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}+\frac {(3 a+b) (a+3 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 a f (a-b)^4 \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 411
Rule 418
Rule 470
Rule 525
Rule 527
Rule 3196
Rubi steps
\begin {align*} \int \frac {\tanh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^{5/2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac {\text {sech}^2(e+f x) \tanh (e+f x)}{3 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {a+(-3 a-2 b) x^2}{\left (1+x^2\right )^{3/2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{3 (a-b) f}\\ &=-\frac {2 (2 a+b) \tanh (e+f x)}{3 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\text {sech}^2(e+f x) \tanh (e+f x)}{3 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {3 a (a+b)-6 b (2 a+b) x^2}{\sqrt {1+x^2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{3 (a-b) (-a+b) f}\\ &=-\frac {b (5 a+3 b) \cosh (e+f x) \sinh (e+f x)}{3 (a-b)^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (2 a+b) \tanh (e+f x)}{3 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\text {sech}^2(e+f x) \tanh (e+f x)}{3 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {3 a^2 (3 a+5 b)-3 a b (5 a+3 b) x^2}{\sqrt {1+x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{9 a (a-b)^2 (-a+b) f}\\ &=-\frac {b (5 a+3 b) \cosh (e+f x) \sinh (e+f x)}{3 (a-b)^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (2 a+b) \tanh (e+f x)}{3 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\text {sech}^2(e+f x) \tanh (e+f x)}{3 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left (8 a b (a+b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 (a-b)^3 (-a+b) f}-\frac {\left ((3 a+b) (a+3 b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 (a-b)^3 (-a+b) f}\\ &=-\frac {b (5 a+3 b) \cosh (e+f x) \sinh (e+f x)}{3 (a-b)^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {8 \sqrt {a} \sqrt {b} (a+b) \cosh (e+f x) E\left (\tan ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 (a-b)^4 f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(3 a+b) (a+3 b) F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a (a-b)^4 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {2 (2 a+b) \tanh (e+f x)}{3 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\text {sech}^2(e+f x) \tanh (e+f x)}{3 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 3.46, size = 252, normalized size = 0.76 \[ -\frac {i \left (2 a b \left (\frac {2 a+b \cosh (2 (e+f x))-b}{a}\right )^{3/2} \left (\left (-5 a^2+2 a b+3 b^2\right ) F\left (i (e+f x)\left |\frac {b}{a}\right .\right )+8 a (a+b) E\left (i (e+f x)\left |\frac {b}{a}\right .\right )\right )-i \sqrt {2} b \left (2 a b (a-b) \sinh (2 (e+f x))+4 b (a+b) \sinh (2 (e+f x)) (2 a+b \cosh (2 (e+f x))-b)+4 (a+b) \tanh (e+f x) (2 a+b \cosh (2 (e+f x))-b)^2-\left ((a-b) \tanh (e+f x) \text {sech}^2(e+f x) (2 a+b \cosh (2 (e+f x))-b)^2\right )\right )\right )}{6 b f (a-b)^4 (2 a+b \cosh (2 (e+f x))-b)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sinh \left (f x + e\right )^{2} + a} \tanh \left (f x + e\right )^{4}}{b^{3} \sinh \left (f x + e\right )^{6} + 3 \, a b^{2} \sinh \left (f x + e\right )^{4} + 3 \, a^{2} b \sinh \left (f x + e\right )^{2} + a^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.40, size = 663, normalized size = 1.99 \[ -\frac {\left (8 \sqrt {-\frac {b}{a}}\, a \,b^{2}+8 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \sinh \left (f x +e \right ) \left (\cosh ^{6}\left (f x +e \right )\right )+\left (13 \sqrt {-\frac {b}{a}}\, a^{2} b -2 \sqrt {-\frac {b}{a}}\, a \,b^{2}-11 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \left (\cosh ^{4}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+\left (4 \sqrt {-\frac {b}{a}}\, a^{3}-6 \sqrt {-\frac {b}{a}}\, a^{2} b +2 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+\left (-\sqrt {-\frac {b}{a}}\, a^{3}+3 \sqrt {-\frac {b}{a}}\, a^{2} b -3 \sqrt {-\frac {b}{a}}\, a \,b^{2}+\sqrt {-\frac {b}{a}}\, b^{3}\right ) \sinh \left (f x +e \right )-\sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, b \left (3 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2}+2 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b -5 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}+8 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b +8 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}\right ) \left (\cosh ^{4}\left (f x +e \right )\right )-\sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \left (3 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{3}-\EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b -7 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}+5 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3}+8 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b -8 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3}\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}{3 \cosh \left (f x +e \right )^{3} \sqrt {-\frac {b}{a}}\, \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (a -b \right )^{4} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh \left (f x + e\right )^{4}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {tanh}\left (e+f\,x\right )}^4}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{4}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________